Volume of a prism

Simple Shapes

The volume of a simple 3d object is equal to the area of the base object multiplied by the height of the object

• Volume of a cube: s³, where s is the side length of the any square side
• Volume of a rectangle: l • w • h
• Volume of a right cylinder: πr²h

Complex Shapes

The shapes above all have right vertices, meaning that their heights are consistently perpendicular. Not all 3D shapes have perpendicular vertices, which means that their volume equations are more complex.

The most likely weird volume equation to be asked about is:

• Volume of a Sphere: 43πr3

Other strange area volumes that can be worth knowing:

• Volume of a Right Cone: πr2h3

Inscribed Volume

One of the more complex volume problems to see on the ACT involves inscribed objects, meaning where one 3D object is inside of another. On the ACT this usually involves a sphere and a circle, either of which can be the interior or exterior object.

The first key to these questions is to draw it out. It will seem complex when it’s first described, but the calculation should become more intuitive once it is drawn out. The objects will have to be described as touching each other at at least one point, but most likely all points. This will then give key information about the relationship between things like the length of the sides of the cube and the radius of the sphere.